Dynamics of Charged and Rotating NUT Black Holes in Rastall Gravity

Freddy P. Zen; Getbogi Hikmawan; Hadyan L. Prihadi; Muhammad F. A. R. Sakti

REVIEW 4 major objections 5 minor 1 cited by

Dynamics of Charged and Rotating NUT Black Holes in Rastall Gravity

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A new Kerr-Newman-NUT black hole solution in Rastall gravity is constructed, and its NUT parameter is shown to shift horizons, thermodynamic stability, and circular orbits.

desk verdict The paper constructs a new Kerr-Newman-NUT-Rastall metric but never checks that it solves the field equations; everything downstream is conditional on an unverified claim. read the letter →

arxiv 1908.09629 v2 pith:PDU26J75 submitted 2019-08-19 gr-qc hep-th

Hadyan L. Prihadi , Muhammad F. A. R. Sakti , Getbogi Hikmawan , Freddy P. Zen This is my paper · ORCID

classification gr-qchep-th

keywords blackholeRastallgravityNUTparameterNewman-JanisalgorithmeventhorizonthermodynamicsequatorialcircularorbitISCO

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to establish that the Kerr-Newman-NUT metric admits a consistent generalization in Rastall gravity, a modified theory in which the energy-momentum tensor is allowed a small non-conservation proportional to the gradient of the Ricci scalar. The authors generate the metric from a static charged quintessence seed using the Newman-Janis algorithm, and they then trace how the NUT parameter and Rastall parameter affect the horizon structure, ergoregion, zero-angular-momentum observers, entropy, temperature, heat capacity, and equatorial circular orbits. Their main dynamical results are that the horizon becomes angle-dependent, that increasing the NUT parameter enlarges the horizon and the static-radius limit and moves the null circular orbit outward, and that the innermost stable circular orbit moves inward as NUT grows. The paper supplies a concrete rotating and twisted black hole solution together with a full orbit and thermodynamics analysis in a modified-gravity setting.

What carries the argument

The load-bearing mechanism is the Newman-Janis algorithm with complex coordinate shifts $\tilde{u}=u-ia\cos\theta+2in\ln\sin\theta$, $\tilde{r}=r+ia\cos\theta-in$, and $\tilde{M}=M+in$, which inject both rotation $a$ and the NUT twist $n$ into the static seed and produce the KNN-R metric in Boyer-Lindquist coordinates. The horizon and ergosurface are controlled by the zeros of $\Delta$ and $g_{tt}$, while the equatorial geodesic analysis is carried by the effective potential $V_{\rm eff}$ derived from the Hamiltonian. The thermodynamic results rest on the slowly rotating horizon area $A_H\simeq 4\pi(r_+^2+n^2)$, from which entropy, temperature, and heat capacity are obtained.

What would settle it

Compute the tensor $G_{\mu\nu}+\kappa\lambda R g_{\mu\nu}-\kappa T_{\mu\nu}$ for the KNN-R metric of Eq. (41) with the electromagnetic-plus-quintessence energy-momentum tensor; if any component fails to vanish identically for general nonzero $\lambda$ and $N_s$, the metric is not a solution of the Einstein-Rastall equations. A cheaper check is to verify whether the complexified energy-momentum tensor still satisfies $\nabla_\mu T^{\mu\nu}=\lambda\nabla^\nu R$ and the symmetry conditions $T^t_t=T^r_r$, $T^\theta_\theta=T^\phi_\phi$ that the static seed required.

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Extended reading notes

Core claim

The paper's central discovery is the Kerr-Newman-NUT-Rastall (KNN-R) solution, its Eq. (41), obtained by applying the Newman-Janis algorithm to the static spherically symmetric charged black hole with quintessential matter in Rastall gravity. The metric has Boyer-Lindquist form with a generalized function $\Delta(r,\theta)=r^2+a^2-n^2-2Mr+Q^2-N_s\Sigma^{(2-\zeta)/2}$, where the extra $N_s$ term encodes the quintessence and Rastall corrections. Because $\Delta$ depends on $\theta$ through $\Sigma=(r^2+(a\cos\theta-n)^2)$, the horizon is not spherical, and it can have inner, outer, and cosmological branches. The paper reports that the NUT parameter $n$ slightly increases the horizon radius, increases the static radius limit for time-like circular orbits, increases the null circular orbit radius, and, in the numerical plots, decreases the innermost stable circular orbit radius. In the slowly rotating limit, it also computes Bekenstein-Hawking entropy, Hawking temperature, and heat capacity, identifying regions of thermodynamic stability as functions of NUT and Rastall parameters.

Load-bearing premise

The whole analysis rests on the assumption that the Newman-Janis algorithm, a complex-coordinate recipe for adding rotation to a static metric, applied to the static charged quintessence seed yields a metric that actually satisfies the Einstein-Rastall field equations; the paper does not substitute the final metric back into those equations to verify this.

Editorial extensions

If this is right

For fixed mass, charge, and rotation, increasing the NUT parameter enlarges the outer horizon, so NUT acts like an additional source of gravitational strength.
The angle-dependent horizon means the black hole silhouette and ergoregion are not spherically symmetric, which would affect shadow and accretion-disk modeling.
Thermodynamic stability regions shift with both the Rastall and NUT parameters, since the heat capacity can change sign as the outer horizon radius varies.
The static radius limit and null circular orbit move outward with NUT, while the innermost stable circular orbit moves inward in the plotted cases, changing the predicted inner edge of an accretion disk.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the KNN-R metric genuinely satisfies the Einstein-Rastall equations, the same Newman-Janis construction could be applied to other static Rastall seeds (for example, different equations of state) to generate a family of rotating NUT solutions; the paper does not attempt this.
The angle-dependent horizon should leave a distinctive imprint on the black hole shadow: a horizon that bulges at the poles would produce a non-circular shadow, a testable prediction the paper does not carry out.
A direct substitution of Eq. (41) into the field equations $G_{\mu\nu}+\kappa\lambda R g_{\mu\nu}=\kappa T_{\mu\nu}$ would settle whether the Newman-Janis procedure preserves the Rastall field equations for non-vacuum sources; the paper does not display that check.
The slowly rotating thermodynamic derivation assumes $a\ll n$ to drop the $\theta$-dependence; extending to arbitrary rotation would require a horizon-averaged temperature and could modify the stability diagram.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Glad you sent this. The paper is a typical NJA construction: take a static charged quintessence black hole in Rastall gravity, run the Newman-Janis algorithm, get a Kerr-Newman-NUT metric, then do a standard tour of horizon, ergosphere, thermodynamics, and geodesics. The specific metric in Eq. (41) is new in the sense that nobody has combined NUT charge with Rastall plus quintessence in one line element, and the orbit analysis is more detailed than in their earlier paper. If the metric is genuine, the paper is a modest but solid contribution to a subfield where this kind of thing is routine.

The problem is the "if." The central claim, that Eq. (41) solves Eq. (2), is never verified. They apply the NJA to a non-vacuum modified-gravity system and simply assert the result is a solution. The complexification rules for the quintessence term, Eqs. (33)-(35), are an ansatz. They never present the rotating electromagnetic potential, never check Maxwell's equations, and never substitute the metric back into the Einstein-Rastall equations. That is a load-bearing omission, not a formal detail.

A concrete symptom is that their Delta depends on theta through Sigma. In consistent Boyer-Lindquist solutions without string-like sources, Delta is a function of r only. The authors acknowledge the theta dependence and try to sell it as physical, but it is more plausible that the coordinate transformation in Eqs. (39)-(40) does not produce an actual solution. Even if the metric does solve the field equations, the paper gives no evidence. The thermodynamics section is also restricted to a << n precisely to drop the theta dependence, so those results do not cover the general metric.

On the positive side, the algebra looks careful as far as it goes. The reduction to Kerr-Newman-NUT when Ns and lambda vanish works, and the geodesic equations are standard. The citation pattern is unobjectionable; Ref. 1 is their own prior static solution, and they lean on it appropriately. There is no fitting-to-data circularity, because nothing is fitted.

Who is this for? People who collect rotating black hole solutions in Rastall gravity. They will want to know whether Eq. (41) actually solves the field equations before using it. As it stands, I would not accept the paper. The fix is straightforward: substitute the metric, compute the Einstein tensor, and show that it matches the appropriate energy-momentum tensor. If they can do that, the paper is citable. If they cannot, the paper is a study of a metric that may not be a black hole in Rastall gravity.

For peer review: I would send it out, but with an instruction to the referee to demand the verification. It is borderline, but the claim is concrete and checkable. I'd rather have a referee confirm it than desk-reject on suspicion.

Referee Report

4 major / 5 minor

Summary. The paper proposes a 'Kerr-Newman-NUT-Rastall' (KNN-R) black hole metric, Eq. (41), constructed by applying the Newman-Janis algorithm to a static, spherically symmetric charged quintessence solution in Rastall gravity. The authors then analyze the resulting geometry: horizons and ergoregions, ZAMO angular velocity, Bekenstein-Hawking thermodynamics under a slowly-rotating approximation, and equatorial circular orbits including the static radius, null circular orbit, and ISCO. The central claim is that Eq. (41) is an exact solution of the Einstein-Rastall field equations, Eq. (2), and that all subsequent dynamical results follow from it.

Significance. If Eq. (41) were a genuine solution, the paper would provide a broad family of rotating charged NUT black holes with quintessence in Rastall gravity, together with a useful catalog of horizon, thermodynamic, and geodesic quantities. The authors are diligent in presenting explicit formulas, tables, and figures, and the static-sector derivation is clearly laid out. However, the central existence claim is not verified: the rotating metric is never substituted back into Eq. (2), the rotating energy-momentum tensor is never defined, and the Newman-Janis algorithm is assumed to preserve the field equations in a non-vacuum modified-gravity setting. The significance of the paper is therefore entirely conditional on a proof that Eq. (41) satisfies Eq. (2), which the manuscript does not supply.

major comments (4)

[Section 2.2, Eq. (41)] The central claim that Eq. (41) solves the Einstein-Rastall equations is not demonstrated. The paper applies the Newman-Janis algorithm to the static solution, but no substitution of Eq. (41) into Eq. (2) is presented, no rotating energy-momentum tensor is written down, and the Maxwell equations for the rotating electromagnetic potential are not checked. The complexification rules in Eqs. (33)-(35) are a prescription inherited from vacuum and electrovacuum general relativity; for a non-vacuum modified-gravity source with an anisotropic quintessence part, they do not automatically generate a solution. This is load-bearing because every later result in Sections 3-5 uses Eq. (41) as its starting point.
[Section 3.1, Eqs. (41)-(42)] The function Delta in Eq. (41) depends on theta through Sigma, so the horizon radius r_+ is a function of theta. In the standard Boyer-Lindquist form of a stationary axisymmetric solution, Delta is a function of r alone; the paper offers only a qualitative statement that the theta dependence arises from surrounding matter and the Rastall parameter, not a derivation. The authors should prove that the surface Delta = 0 is a null hypersurface and that the standard horizon thermodynamics apply to it; otherwise the area and temperature formulas in Section 4 are not justified.
[Section 4, Eqs. (49)-(57)] The thermodynamic calculation assumes a slowly rotating black hole whose horizon is effectively theta-independent and spherical. However, Eq. (42) gives r_+(theta) whenever N_s is nonzero, and the approximation a << n still retains terms linear in a through the cross term -2an cos(theta) in Sigma. Dropping the theta dependence would require setting a = 0 exactly, or at least a separate justification that the linear terms in a are negligible for the horizon locus and the area element. The area integral in Eq. (49) and the surface gravity in Eq. (52) should be evaluated on the actual surface r = r_+(theta), not on a spherical surface, unless this is explicitly proven.
[Section 5, Eqs. (65)-(75)] The equatorial circular-orbit analysis imposes theta = pi/2 and dot-theta = 0, but it does not verify that dV_eff/dtheta = 0 at theta = pi/2. For a metric with a NUT parameter and theta-dependent metric functions, this condition is not automatic; if it fails, the orbits in Tables 3 and 4 and Figure 8 are not true equatorial circular geodesics. The paper should either prove that the equatorial plane is totally geodesic for the KNN-R metric or account for the theta-direction force in the effective potential.

minor comments (5)

[Section 2.1, Eqs. (16)-(17)] The exponent zeta and the quintessence term are written as N_s/r^zeta and later as N_s/Sigma^{zeta/2}; the relation between these expressions and the definition of zeta in Eq. (17) should be stated explicitly to avoid confusion.
[Section 3, Tables 2 and 3] Some numerical entries in the tables appear suspiciously repeated, for example the values for n = 0.4 and n = 0.6 in Table 3 at kappa-lambda = 1/10; please verify the computations and the precision reported.
[Section 5.2, Eq. (73)] Equation (73) is extremely long and effectively unreadable in the main text; moving the derivation to an appendix and presenting the final condition in a more compact form would improve clarity.
[Abstract and Section 5.1] The term 'static radius limit' is used in the abstract and introduction but is only defined in Section 5.1; define it at first use.
[Throughout] There are numerous minor language and typographical issues, such as 'the dependence of the other ... contained explicitly' and inconsistent use of notation for the Rastall parameter; a careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the KNN-R metric is an explicit Newman-Janis ansatz, and the horizon, thermodynamic, and geodesic results are algebraic consequences of that metric; the unverified field-equation check is a correctness gap, not a circular reduction.

full rationale

Walking the derivation chain, the static seed solution Eq. (16) is obtained by inserting the stated ansatz f(r) = 1 - 2M/r + Q^2/r^2 + h(r) into the Einstein-Rastall field equations (13)-(14) and solving for h(r) and the exponents ζ and Ws. These quantities are algebraic outputs of the differential equations, not fitted to the later horizon, temperature, or ISCO results. The rotating and NUT metric is then generated by an explicit Newman-Janis procedure: the complexification rules (33)-(35) are stated, the transformed function f~(r,theta) is given in Eq. (36), and the final Boyer-Lindquist line element is written as Eq. (41). All subsequent quantities — horizons from Δ = 0, ergosurface from g_tt = 0, ZAMO angular velocity, surface gravity, temperature, heat capacity, static radius, null circular orbit, and ISCO — are computed directly from that line element and the geodesic Lagrangian. None of these later quantities is reinserted into the construction to fix a free parameter, and no predicted result is used to define the input metric. The only overlapping-author citation is Ref. 1, used for the quintessence energy-momentum structure and as the static solution to be generalized; however, Section 2.1 re-derives the static solution and the Rastall-dependent exponents from Eqs. (13)-(14) in the present paper, so the self-citation is corroborative rather than load-bearing. The genuine weakness is that Eq. (41) is never substituted back into Eq. (2), and no rotating effective stress-energy tensor or Maxwell potential for the NJA output is presented; but that is an unverified correctness assumption, not a circularity. An unsupported assertion that the Newman-Janis algorithm produces a solution is not the same as a derivation whose conclusion is assumed among its inputs. Therefore no circular step is identified.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central solution is constructed from the authors' prior static solution (Ref. 1) via the Newman-Janis algorithm. It inherits the matter model assumptions and adds the unverified NJA validity assumption. No new particles or forces are introduced.

free parameters (4)

kappa lambda (Rastall parameter) = not fitted; varied (0, 1/10, 1/6, 1/4)
Coupling constant measuring deviation from Einstein gravity. No observational constraint; chosen for illustration. Restriction 0 <= kappa lambda <= 1/4 imposed to keep quintessence density non-negative.
Ns (quintessence density) = not fitted; e.g., 0.01, -0.01
Integration constant of the matter field. Chosen for plots; no observational constraint.
omega (equation-of-state parameter) = not fitted; 0, 1/3, -1/3, -2/3, -1
Equation of state of the surrounding field. Chosen to represent dust, radiation, quintessence, and cosmological constant.
n (NUT parameter) = not fitted; varied (0 to 1)
NUT charge parameter studied throughout the paper. No observational constraint; chosen for illustration.

assumptions (6)

domain assumption The Einstein-Rastall field equation G_mu_nu + kappa lambda R g_mu_nu = kappa T_mu_nu (Eq. 2) is the correct equation of motion.
The paper adopts Rastall's modification of general relativity without independent evidence or comparison with observations.
domain assumption The total energy-momentum tensor is the sum of the electromagnetic and quintessence parts with the forms given in Eqs. (11) and (12).
The forms are chosen to satisfy the symmetry conditions H_t^t = H_r^r and H_theta^theta = H_phi^phi; they are not derived from a more fundamental theory.
domain assumption The static metric function takes the ansatz f(r) = 1 - 2M/r + Q^2/r^2 - Ns/r^zeta with zeta given by Eq. (17).
The ansatz is chosen to recover Reissner-Nordstrom and Kiselev limits; it is not a unique solution.
ad hoc to paper The Newman-Janis algorithm applied to a static solution yields a valid solution of the Einstein-Rastall field equations.
The paper relies on NJA without verifying the generated metric satisfies Eq. (2); in non-vacuum modified gravity this is not guaranteed. This is the weakest assumption.
domain assumption For thermodynamics, the slowly-rotating limit a << n is valid and removes theta dependence from the horizon.
Invoked in Section 4 to justify using a constant-r+ horizon area; no systematic error estimate is provided.
domain assumption The Bekenstein-Hawking area-entropy law S = A/4 applies in Rastall gravity.
Used without derivation or discussion of possible corrections in modified gravity.

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Pith. "Pith review of Dynamics of Charged and Rotating NUT Black Holes in Rastall Gravity." pith.science (2026). https://pith.science/paper/PDU26J75

@misc{pith2026190809629,
  author       = {Pith},
  title        = {Pith review of: Dynamics of Charged and Rotating NUT Black Holes in Rastall Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PDU26J75}},
  note         = {Machine review of arXiv:1908.09629}
}

read the original abstract

In this work, we generalized the Kerr-Newman-NUT black hole solution in Rastall gravity from Ref. 1. Here we are more focused on the black hole dynamics such as the event horizons, ergosurface, ZAMO, thermodynamic properties, and the equatorial circular orbit around the black hole such as static radius limit, null equatorial circular orbit, and innermost stable circular orbit. We present how the NUT and Rastall parameter affects the dynamic of the black hole.

Figures

Figures reproduced from arXiv: 1908.09629 by the authors.

**Figure 2.** Plot showing the variation of ∆ function with respect [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** Plot showing the structure of the horizon and ergosur [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: Plot showing the temperature of a KNN-R black hole hor [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]

**Figure 5.** Figure 5: Plot showing the heat capacity of a KNN-R black hole ho [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]

**Figure 6.** Figure 6: It is shown that the NUT parameter gives us an important info [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]

**Figure 6.** Figure 6: Plot of the co-rotating energy E+ and angular momentum Lz+ with respect to r for a KNN-R black hole surrounded by dust field with M = 1, a = 0.9, Q = 0.5, κλ = 0, and Ns = −0.01. For the same amount of energy and angular momentum, a time-like particle could revolve clo…

**Figure 7.** Figure 7: Plot of the counter-rotating energy E− and angular momentum Lz− with respect to r for a KNN-R black hole surrounded by dust field with M = 1, a = 0.9, Q = 0.5, κλ = 0, and Ns = −0.01. This gives a similar result with the previous co-rotating plot. Interestingly, Eq. (7…

**Figure 8.** Figure 8: Plot showing the behavior of rISCO when the NUT parameter n varies for different value of Rastall parameter for a KNN-R black hole surrounded by dust (left) and quintessence (right) field. Here blue-dot, black-dot-dashed, orange-dashed, and red-solid line represents κλ…

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

M. F. A. R. Sakti, A. Suroso, F. P. Zen. Kerr-Newman-NUT bla ck hole with quintessen- tial matter in Rastall theory of gravity. arXiv preprint gr- qc/1901.09163 (2019)

arXiv 2019

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P. Rastall. Generalization of the Einstein Theory. Phys. Rev. D 6, (1972) 3357

work page 1972

[3] [3]

E. J. Copeland, M. Sami, and S. Tsujikawa. Dynamics of dark energy. Int. J. Mod. Phys. D 15, (2006) 1753

work page 2006

[4] [4]

V. V. Kiselev. Quintessence and black holes. Class. Quant. Grav. 20, (2003) 1187-1198

work page 2003

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S. G. Ghosh. Rotating black hole and quintessence. Eur. Phys. J. C 76, (2016) 222

work page 2016

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Toshmatov, Z

B. Toshmatov, Z. Stuchl, dan B. Ahmedov. Rotating black ho le solutions with quintessential energy. Eur. Phys. J. Plus 98, (2017) 132

work page 2017

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